B-coloring of $K_{2,t}$-free planar graphs

A B-coloring of a graph $G$ is a proper edge-coloring in which every $4$-cycle receives four distinct colors; let $q_B(G)$ be the minimum number of colors in such a coloring. Every graph of maximum degree $\mathrm{\Delta }$ is $K_{2,\mathrm{\Delta }+1}$-free; hence the known $2\mathrm{\Delta }$ bound for planar graphs with $\mathrm{\Delta }\ge 38$ (Kong et al., 2026) motivates our study of $K_{2,t}$-free planar graphs, where $t\ge 2$ is an integer. We prove $q_B(G)=\mathrm{\Delta }(G)$ when $t=2$ and $\mathrm{\Delta }(G)\ge 7$, or when $t\ge 3$ and $\mathrm{\Delta }(G)\ge 14(t-1)$. For $t\ge 35$, the bound $q_B(G)\le \mathrm{\Delta }(G)+t-1$ holds regardless of $\mathrm{\Delta }(G)$; for every $t\ge 2$, it also holds when $\mathrm{\Delta }(G)>428$. Finally, for every integer $k\ge 1$, every $k$-degenerate $K_{2,t}$-free graph satisfies $q_B(G)\le \mathrm{\Delta }(G)+(k-1)\min \{t-1,\mathrm{\Delta }(G)\}$, with equality for $K_{k,t-1}$ when $k\ge 2$ and $t-1\ge k$.